What is a Graph Cycle? | Graph Theory, Cycles, Cyclic Graphs, Simple Cycles
What is a graph cycle? In graph theory, a cycle is a way of moving through a graph. We can think of a cycle as being a sequence of vertices in a graph, such that consecutive vertices are adjacent, and all vertices are distinct except for the first and last vertex, which are required to be the same.
A cycle can be represented as a sequence of vertices in a graph, but it is sometimes also represented in other ways. For example, if ( a, b, c, d, a ) is a cycle in a graph, it could also be represented as a sequence of edges. We could write it as ( ab, bc, cd, da ). Additionally, a cycle is sometimes written as an alternating sequence of vertices and edges like this: ( a, ab, b, bc, c, cd, d, da, a ). These are just a few ways to write cycles.
The length of a cycle is the number of edges traversed in the cycle. So if a cycle is written as a sequence of vertices, the length of the cycle is one less than the number of vertices in the sequence. If the cycle is written as a sequence of edges then the length is just the number of edges in the sequence. The length of a cycle must be 3 or greater.
Sometimes cycles are defined slightly differently, so that vertices are allowed to be repeated but not edges. In this case, the cycles we previously discussed, that don't repeat vertices except for the first and last, are often called simple cycles.
I hope you find this video helpful, and be sure to ask any questions down in the comments!
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